Differential Geometry textbooks for someone interested in Algebraic Geometry

Solution 1:

A very good,concise and completely modern text on differential geometry is Gerard Walschap's Metric Structures In Differential Geometry.

What you're looking for is a relatively short, very readable differential geometry book with broad coverage,modern language and minimal prerequisites. Such a book by necessity would shift most of the results to the exercises and therefore would create a text the graduate student would need to learn actively with. Not only would this allow the student to learn the subject quickly, it would help acclimate them to taking the training wheels off in graduate school without being too discouragingly dense. It would need to have clear definitions, sections broken into bite-size pieces and a couple of well-placed diagrams wouldn't hurt.

If that's what you're looking for, you can't do better then Walschap. In a mere 226 pages, Walschap races the student through all the major broad strokes of the subject without making their eyes glaze over-quite a feat. He covers differentiable manifolds,multilinear algebra and forms,vector and fiber bundles,homotopy groups over spheres (a tough topic without algebraic topology, but Walschap does a good job covering just the bare bones), connection structures on bundles such as Riemannian structures and the book finishes with an elementary introduction to complex differential geometry and characteristic classes.The book has a definite topological bent by emphasizing fiber bundles rather then vector bundles. Walschap tries very hard to keep the prerequisites to a minimum: a good grasp of real analysis,point set topology and algebra. The author introduces algebraic topology only when it's needed. For example the fundamental group is introduced as a special case of homotopy groups.Another example is that cohomology isn't used in characteristic classes, they are constructed directly using the Weil homomorphism. This is more old fashioned and involved algebraically, but conceptually simpler.

The language of the book is completely modern, commutative diagrams are used throughout. The real joy of the book is the hundreds of integrated exercises-they're all substantial and none are too hard. A great deal of the material is developed in these exercises,so the student really needs to work through them. But working through them is half the fun-and the fact Walschap makes working exercises fun is a measure of his skill as a teacher.I strongly advise you give this wonderful and unorthodox text a look.

Solution 2:

How about Manifolds, sheaves, and cohomology by Wedhorn (you can buy this as a book, same title as the notes)? While it does not give a treatment of deep topics in differential geometry, it does define manifolds as ringed spaces.

Solution 3:

If you want something emphasizing the algebraic mathematical structures at large underpinning differential geometry, something modern, abstract and conceptual, then "Natural Operations in Differential Geometry" by Ivan Kolár, Peter W. Michor and Jan Slovák is the thing for you! I warn you, though, that after some contact with it you might want to come back to a more traditional presentation, because your visual intuition will suffer greatly! On the other hand, if you have a thing for schemes, then maybe you'll feel comfortable with this one too.